SDRGVX checks the nonsymmetric generalized eigenvalue problem


 expert driver SGGEVX.


 SGGEVX computes the generalized eigenvalues, (optionally) the left


 and/or right eigenvectors, (optionally) computes a balancing


 transformation to improve the conditioning, and (optionally)


 reciprocal condition numbers for the eigenvalues and eigenvectors.


 When SDRGVX is called with NSIZE > 0, two types of test matrix pairs


 are generated by the subroutine SLATM6 and test the driver SGGEVX.


 The test matrices have the known exact condition numbers for


 eigenvalues. For the condition numbers of the eigenvectors


 corresponding the first and last eigenvalues are also know


 ``exactly'' (see SLATM6).


 For each matrix pair, the following tests will be performed and


 compared with the threshold THRESH.


 (1) max over all left eigenvalue/-vector pairs (beta/alpha,l) of


    | l**H * (beta A - alpha B) | / ( ulp max( |beta A|, |alpha B| ) )


     where l**H is the conjugate transpose of l.


 (2) max over all right eigenvalue/-vector pairs (beta/alpha,r) of


       | (beta A - alpha B) r | / ( ulp max( |beta A|, |alpha B| ) )


 (3) The condition number S(i) of eigenvalues computed by SGGEVX


     differs less than a factor THRESH from the exact S(i) (see


     SLATM6).


 (4) DIF(i) computed by STGSNA differs less than a factor 10*THRESH


     from the exact value (for the 1st and 5th vectors only).


 Test Matrices


 =============


 Two kinds of test matrix pairs


          (A, B) = inverse(YH) * (Da, Db) * inverse(X)


 are used in the tests:


 1: Da = 1+a   0    0    0    0    Db = 1   0   0   0   0


          0   2+a   0    0    0         0   1   0   0   0


          0    0   3+a   0    0         0   0   1   0   0


          0    0    0   4+a   0         0   0   0   1   0


          0    0    0    0   5+a ,      0   0   0   0   1 , and


 2: Da =  1   -1    0    0    0    Db = 1   0   0   0   0


          1    1    0    0    0         0   1   0   0   0


          0    0    1    0    0         0   0   1   0   0


          0    0    0   1+a  1+b        0   0   0   1   0


          0    0    0  -1-b  1+a ,      0   0   0   0   1 .


 In both cases the same inverse(YH) and inverse(X) are used to compute


 (A, B), giving the exact eigenvectors to (A,B) as (YH, X):


 YH:  =  1    0   -y    y   -y    X =  1   0  -x  -x   x


         0    1   -y    y   -y         0   1   x  -x  -x


         0    0    1    0    0         0   0   1   0   0


         0    0    0    1    0         0   0   0   1   0


         0    0    0    0    1,        0   0   0   0   1 , where


 a, b, x and y will have all values independently of each other from


 { sqrt(sqrt(ULP)),  0.1,  1,  10,  1/sqrt(sqrt(ULP)) }.

NSIZE

          NSIZE is INTEGER


          The number of sizes of matrices to use.  NSIZE must be at


          least zero. If it is zero, no randomly generated matrices


          are tested, but any test matrices read from NIN will be


          tested.

THRESH

          THRESH is REAL


          A test will count as 'failed' if the 'error', computed as


          described above, exceeds THRESH.  Note that the error


          is scaled to be O(1), so THRESH should be a reasonably


          small multiple of 1, e.g., 10 or 100.  In particular,


          it should not depend on the precision (single vs. double)


          or the size of the matrix.  It must be at least zero.

NIN

          NIN is INTEGER


          The FORTRAN unit number for reading in the data file of


          problems to solve.

NOUT

          NOUT is INTEGER


          The FORTRAN unit number for printing out error messages


          (e.g., if a routine returns IINFO not equal to 0.)

A

          A is REAL array, dimension (LDA, NSIZE)


          Used to hold the matrix whose eigenvalues are to be


          computed.  On exit, A contains the last matrix actually used.

LDA

          LDA is INTEGER


          The leading dimension of A, B, AI, BI, Ao, and Bo.


          It must be at least 1 and at least NSIZE.

B

          B is REAL array, dimension (LDA, NSIZE)


          Used to hold the matrix whose eigenvalues are to be


          computed.  On exit, B contains the last matrix actually used.

AI

          AI is REAL array, dimension (LDA, NSIZE)


          Copy of A, modified by SGGEVX.

BI

          BI is REAL array, dimension (LDA, NSIZE)


          Copy of B, modified by SGGEVX.

ALPHAR

          ALPHAR is REAL array, dimension (NSIZE)

ALPHAI

          ALPHAI is REAL array, dimension (NSIZE)

BETA

          BETA is REAL array, dimension (NSIZE)


          On exit, (ALPHAR + ALPHAI*i)/BETA are the eigenvalues.

VL

          VL is REAL array, dimension (LDA, NSIZE)


          VL holds the left eigenvectors computed by SGGEVX.

VR

          VR is REAL array, dimension (LDA, NSIZE)


          VR holds the right eigenvectors computed by SGGEVX.

ILO

                ILO is INTEGER

IHI

                IHI is INTEGER

LSCALE

                LSCALE is REAL array, dimension (N)

RSCALE

                RSCALE is REAL array, dimension (N)

S

                S is REAL array, dimension (N)

STRU

                STRU is REAL array, dimension (N)

DIF

                DIF is REAL array, dimension (N)

DIFTRU

                DIFTRU is REAL array, dimension (N)

WORK

          WORK is REAL array, dimension (LWORK)

LWORK

          LWORK is INTEGER


          Leading dimension of WORK.  LWORK >= 2*N*N+12*N+16.

IWORK

          IWORK is INTEGER array, dimension (LIWORK)

LIWORK

          LIWORK is INTEGER


          Leading dimension of IWORK.  Must be at least N+6.

RESULT

                RESULT is REAL array, dimension (4)

BWORK

          BWORK is LOGICAL array, dimension (N)

INFO

          INFO is INTEGER


          = 0:  successful exit


          < 0:  if INFO = -i, the i-th argument had an illegal value.


          > 0:  A routine returned an error code.

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